On full cover property of ordered fields. (English) Zbl 0823.12006

Šalát, Tibor

Math. Slovaca 44, No. 5, 501-504 (1994).

The author defines the notion of a full cover of an interval \([a,b]\) in an ordered field \(A\) as follows: A class \(S\) of intervals \(I\subseteq [a,b]\) is said to be a full cover of \([a, b]\) if \(\forall x\in [a,b]\) \(\exists \delta (x)> 0_ A\) in \(A\) such that \(S\) contains each closed interval \(I\subseteq [a,b]\) of length less than \(\delta (x)\) and containing \(x\). Moreover, he defines that an ordered field \(A\) has the full cover property (FC-property) provided that for each \(a,b\in A\), \(a<b\), the following statement holds: If \(S\) is a full cover of \([a,b]\), then there exists a partition \(a_ 0= x_ 0< x_ 1< \cdots < x_ m= b\), \(x_ j\in A\), such that \(| x_{k-1}, x_ k|\in S\), \(k=1,\dots, m\). Then he proves that an Archimedean ordered field \(A\) is complete iff it has the FC-property. He gives some applications of this property.

Reviewer: J.Močkoř (Ostrava)

MSC:

12J15	Ordered fields
12J10	Valued fields

Keywords:

FC-property; full cover of an interval; ordered field; Archimedean ordered field

References:

[1]	ALEXANDROV P. S.: Introduction to General Theory of Sets and Functions. (Czech translation), NCSAV, Prague, 1954.
[2]	BOTSKO M. W.: A unified treatment of various theorems in elementary analysis. Amer. Math. Monthly 94 (1987), 450-452.
[3]	BOTSKO M. W.: The use of full covers in real analysis. Amer. Math. Monthly 96 (1989), 328-333. · Zbl 0721.26003
[4]	COHEN L. W.-EHRLICH G.: The Structure of Real Number System. D. Van Nostrad-Princeton-New Jersey, Toronto-New York-London, 1963. · Zbl 0118.28501
[5]	ŠALÁT T.-TARABOVÁ A.: On properties of Archimedean ordered fields that are equivalent to the completeness. Acta Math. Univ. Comenian. XLII-XLIII (1983), 299-306. · Zbl 0598.12026
[6]	SIKORSKI R.: Real Functions I. (Polish), PWN, Warshaw, 1958.
[7]	THOMSON B. S.: On full covering properties. Real Anal. Exchange 6 (1980-81), 77-91. · Zbl 0459.26004